/*
 *   This program is free software: you can redistribute it and/or modify
 *   it under the terms of the GNU General Public License as published by
 *   the Free Software Foundation, either version 3 of the License, or
 *   (at your option) any later version.
 *
 *   This program is distributed in the hope that it will be useful,
 *   but WITHOUT ANY WARRANTY; without even the implied warranty of
 *   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 *   GNU General Public License for more details.
 *
 *   You should have received a copy of the GNU General Public License
 *   along with this program.  If not, see <http://www.gnu.org/licenses/>.
 */

/*
 *    Optimization.java
 *    Copyright (C) 2003-2012 University of Waikato, Hamilton, New Zealand
 *
 */

package weka.core;

import weka.core.TechnicalInformation.Field;
import weka.core.TechnicalInformation.Type;

import weka.core.matrix.Matrix;

/**
 * Implementation of Active-sets method with BFGS update to solve optimization
 * problem with only bounds constraints in multi-dimensions.  In this
 * implementation we consider both the lower and higher bound constraints. <p/>
 *
 * Here is the sketch of our searching strategy, and the detailed description
 * of the algorithm can be found in the Appendix of Xin Xu's MSc thesis:<p/>
 * Initialize everything, incl. initial value, direction, etc.<p/>
 * LOOP (main algorithm):<br/>
 * 
 * 1. Perform the line search using the directions for free variables<br/>
 * 1.1  Check all the bounds that are not "active" (i.e. binding variables)
 *      and compute the feasible step length to the bound for each of them<br/>
 * 1.2  Pick up the least feasible step length, say \alpha, and set it as 
 *      the upper bound of the current step length, i.e.
 *      0&lt;\lambda&lt;=\alpha<br/>
 * 1.3  Search for any possible step length&lt;=\alpha that can result the 
 *      "sufficient function decrease" (\alpha condition) AND "positive
 *      definite inverse Hessian" (\beta condition), if possible, using
 *      SAFEGUARDED polynomial interpolation.  This step length is "safe" and
 *      thus is used to compute the next value of the free variables .<br/>
 * 1.4  Fix the variable(s) that are newly bound to its constraint(s).<p/>     
 *
 * 2. Check whether there is convergence of all variables or their gradients.
 *    If there is, check the possibilities to release any current bindings of
 *    the fixed variables to their bounds based on the "reliable" second-order
 *    Lagarange multipliers if available.  If it's available and negative for
 *    one variable, then release it.  If not available, use first-order
 *    Lagarange multiplier to test release.  If there is any released
 *    variables, STOP the loop.  Otherwise update the inverse of Hessian matrix
 *    and gradient for the newly released variables and CONTINUE LOOP.<p/>
 *
 * 3. Use BFGS formula to update the inverse of Hessian matrix.  Note the 
 *    already-fixed variables must have zeros in the corresponding entries
 *    in the inverse Hessian.<p/>  
 *
 * 4. Compute the new (newton) search direction d=H^{-1}*g, where H^{-1} is the 
 *    inverse Hessian and g is the Jacobian.  Note that again, the already-
 *    fixed variables will have zero direction.<p/>
 *
 * ENDLOOP<p/>
 *
 * A typical usage of this class is to create your own subclass of this class
 * and provide the objective function and gradients as follows:<p/>
 * <pre>
 * class MyOpt extends Optimization {
 *   // Provide the objective function
 *   protected double objectiveFunction(double[] x) {
 *       // How to calculate your objective function...
 *       // ...
 *   }
 *
 *   // Provide the first derivatives
 *   protected double[] evaluateGradient(double[] x) {
 *       // How to calculate the gradient of the objective function...
 *       // ...
 *   }
 *
 *   // If possible, provide the index^{th} row of the Hessian matrix
 *   protected double[] evaluateHessian(double[] x, int index) {
 *      // How to calculate the index^th variable's second derivative
 *      // ... 
 *   }
 * }
 * </pre>
 *
 * When it's the time to use it, in some routine(s) of other class...
 * <pre>
 * MyOpt opt = new MyOpt();
 * 
 * // Set up initial variable values and bound constraints
 * double[] x = new double[numVariables];
 * // Lower and upper bounds: 1st row is lower bounds, 2nd is upper
 * double[] constraints = new double[2][numVariables];
 * ...
 *
 * // Find the minimum, 200 iterations as default
 * x = opt.findArgmin(x, constraints); 
 * while(x == null){  // 200 iterations are not enough
 *    x = opt.getVarbValues();  // Try another 200 iterations
 *    x = opt.findArgmin(x, constraints);
 * }
 *
 * // The minimal function value
 * double minFunction = opt.getMinFunction();
 * ...
 * </pre>
 * 
 * It is recommended that Hessian values be provided so that the second-order
 * Lagrangian multiplier estimate can be calcluated.  However, if it is not
 * provided, there is no need to override the <code>evaluateHessian()</code>
 * function.<p/>
 *
 * REFERENCES (see also the <code>getTechnicalInformation()</code> method):<br/>
 * The whole model algorithm is adapted from Chapter 5 and other related
 * chapters in Gill, Murray and Wright(1981) "Practical Optimization", Academic
 * Press.  and Gill and Murray(1976) "Minimization Subject to Bounds on the
 * Variables", NPL Report NAC72, while Chong and Zak(1996) "An Introduction to
 * Optimization", John Wiley &amp; Sons, Inc. provides us a brief but helpful
 * introduction to the method. <p/>
 *
 * Dennis and Schnabel(1983) "Numerical Methods for Unconstrained Optimization
 * and Nonlinear Equations", Prentice-Hall Inc. and Press et al.(1992) "Numeric
 * Recipe in C", Second Edition, Cambridge University Press. are consulted for
 * the polynomial interpolation used in the line search implementation.  <p/>
 *
 * The Hessian modification in BFGS update uses Cholesky factorization and two
 * rank-one modifications:<br/>
 * Bk+1 = Bk + (Gk*Gk')/(Gk'Dk) + (dGk*(dGk)'))/[alpha*(dGk)'*Dk]. <br/>
 * where Gk is the gradient vector, Dk is the direction vector and alpha is the
 * step rate. <br/>
 * This method is due to Gill, Golub, Murray and Saunders(1974) ``Methods for
 * Modifying Matrix Factorizations'', Mathematics of Computation, Vol.28,
 * No.126, pp 505-535. <p/>
 *
 * @author Xin Xu (xx5@cs.waikato.ac.nz)
 * @version $Revision: 8076 $ 
 * @see #getTechnicalInformation()
 */
public abstract class Optimization
    implements TechnicalInformationHandler, RevisionHandler {
    
    protected double m_ALF = 1.0e-4;

    protected double m_BETA = 0.9;    

    protected double m_TOLX = 1.0e-6;
   
    protected double m_STPMX = 100.0;
    
    protected int m_MAXITS = 200;
    
    protected static boolean m_Debug = false;
    
    /** function value */
    protected double m_f;    
 
    /** G'*p */
    private double m_Slope;
    
    /** Test if zero step in lnsrch */
    protected boolean m_IsZeroStep = false;
    
    /** Used when iteration overflow occurs */
    protected double[] m_X;
    
    /** Compute machine precision */
    protected static double m_Epsilon, m_Zero; 
    static {
	m_Epsilon=1.0;
	while(1.0+m_Epsilon > 1.0){
	    m_Epsilon /= 2.0;	    
	}
	m_Epsilon *= 2.0;
	m_Zero = Math.sqrt(m_Epsilon);
	if (m_Debug)
	    System.err.print("Machine precision is "+m_Epsilon+
			     " and zero set to "+m_Zero);
    }
    
    /**
     * Returns an instance of a TechnicalInformation object, containing 
     * detailed information about the technical background of this class,
     * e.g., paper reference or book this class is based on.
     * 
     * @return the technical information about this class
     */
    public TechnicalInformation getTechnicalInformation() {
      TechnicalInformation 	result;
      TechnicalInformation 	additional;
      
      result = new TechnicalInformation(Type.MASTERSTHESIS);
      result.setValue(Field.AUTHOR, "Xin Xu");
      result.setValue(Field.YEAR, "2003");
      result.setValue(Field.TITLE, "Statistical learning in multiple instance problem");
      result.setValue(Field.SCHOOL, "University of Waikato");
      result.setValue(Field.ADDRESS, "Hamilton, NZ");
      result.setValue(Field.NOTE, "0657.594");

      additional = result.add(Type.BOOK);
      additional.setValue(Field.AUTHOR, "P. E. Gill and W. Murray and M. H. Wright");
      additional.setValue(Field.YEAR, "1981");
      additional.setValue(Field.TITLE, "Practical Optimization");
      additional.setValue(Field.PUBLISHER, "Academic Press");
      additional.setValue(Field.ADDRESS, "London and New York");
      
      additional = result.add(Type.TECHREPORT);
      additional.setValue(Field.AUTHOR, "P. E. Gill and W. Murray");
      additional.setValue(Field.YEAR, "1976");
      additional.setValue(Field.TITLE, "Minimization subject to bounds on the variables");
      additional.setValue(Field.INSTITUTION, "National Physical Laboratory");
      additional.setValue(Field.NUMBER, "NAC 72");
      
      additional = result.add(Type.BOOK);
      additional.setValue(Field.AUTHOR, "E. K. P. Chong and S. H. Zak");
      additional.setValue(Field.YEAR, "1996");
      additional.setValue(Field.TITLE, "An Introduction to Optimization");
      additional.setValue(Field.PUBLISHER, "John Wiley and Sons");
      additional.setValue(Field.ADDRESS, "New York");
      
      additional = result.add(Type.BOOK);
      additional.setValue(Field.AUTHOR, "J. E. Dennis and R. B. Schnabel");
      additional.setValue(Field.YEAR, "1983");
      additional.setValue(Field.TITLE, "Numerical Methods for Unconstrained Optimization and Nonlinear Equations");
      additional.setValue(Field.PUBLISHER, "Prentice-Hall");
      
      additional = result.add(Type.BOOK);
      additional.setValue(Field.AUTHOR, "W. H. Press and B. P. Flannery and S. A. Teukolsky and W. T. Vetterling");
      additional.setValue(Field.YEAR, "1992");
      additional.setValue(Field.TITLE, "Numerical Recipes in C");
      additional.setValue(Field.PUBLISHER, "Cambridge University Press");
      additional.setValue(Field.EDITION, "Second");
      
      additional = result.add(Type.ARTICLE);
      additional.setValue(Field.AUTHOR, "P. E. Gill and G. H. Golub and W. Murray and M. A. Saunders");
      additional.setValue(Field.YEAR, "1974");
      additional.setValue(Field.TITLE, "Methods for modifying matrix factorizations");
      additional.setValue(Field.JOURNAL, "Mathematics of Computation");
      additional.setValue(Field.VOLUME, "28");
      additional.setValue(Field.NUMBER, "126");
      additional.setValue(Field.PAGES, "505-535");
      
      return result;
    }
    
    /**
     * Subclass should implement this procedure to evaluate objective
     * function to be minimized
     * 
     * @param x the variable values
     * @return the objective function value
     * @throws Exception if something goes wrong
     */
    protected abstract double objectiveFunction(double[] x) throws Exception;

    /**
     * Subclass should implement this procedure to evaluate gradient
     * of the objective function
     * 
     * @param x the variable values
     * @return the gradient vector
     * @throws Exception if something goes wrong
     */
    protected abstract double[] evaluateGradient(double[] x) throws Exception;

    /**
     * Subclass is recommended to override this procedure to evaluate second-order
     * gradient of the objective function.  If it's not provided, it returns
     * null.
     *
     * @param x the variables
     * @param index the row index in the Hessian matrix
     * @return one row (the row #index) of the Hessian matrix, null as default
     * @throws Exception if something goes wrong
     */
    protected double[] evaluateHessian(double[] x, int index) throws Exception{
	return null;
    }

    /**
     * Get the minimal function value
     *
     * @return minimal function value found
     */
    public double getMinFunction() {
      return m_f;
    }

    /**
     * Set the maximal number of iterations in searching (Default 200)
     *
     * @param it the maximal number of iterations
     */
    public void setMaxIteration(int it) {
      m_MAXITS=it;
    }
      
    /**
     * Set whether in debug mode
     *
     * @param db use debug or not
     */
    public void setDebug(boolean db) {
      m_Debug = db;
    }
    
    /**
     * Get the variable values.  Only needed when iterations exceeds 
     * the max threshold.
     *
     * @return the current variable values
     */
    public double[] getVarbValues() {
      return m_X;
    }
    
    /**
     * Find a new point x in the direction p from a point xold at which the
     * value of the function has decreased sufficiently, the positive 
     * definiteness of B matrix (approximation of the inverse of the Hessian)
     * is preserved and no bound constraints are violated.  Details see "Numerical 
     * Methods for Unconstrained Optimization and Nonlinear Equations".
     * "Numeric Recipes in C" was also consulted.
     *
     * @param xold old x value 
     * @param gradient gradient at that point
     * @param direct direction vector
     * @param stpmax maximum step length
     * @param isFixed indicating whether a variable has been fixed
     * @param nwsBounds non-working set bounds.  Means these variables are free and
     *                  subject to the bound constraints in this step
     * @param wsBdsIndx index of variables that has working-set bounds.  Means
     *                  these variables are already fixed and no longer subject to
     *                  the constraints
     * @return new value along direction p from xold, null if no step was taken
     * @throws Exception if an error occurs
     */
    public double[] lnsrch(double[] xold, double[] gradient, 
			   double[] direct, double stpmax,
			   boolean[] isFixed, double[][] nwsBounds,
			   DynamicIntArray wsBdsIndx)
	throws Exception {
	
	int i, k,len=xold.length, 
	    fixedOne=-1; // idx of variable to be fixed
	double alam, alamin; // lambda to be found, and its lower bound
	
	// For convergence and bound test
	double temp,test,alpha=Double.POSITIVE_INFINITY,fold=m_f,sum; 
	
	// For cubic interpolation
	double a,alam2=0,b,disc=0,maxalam=1.0,rhs1,rhs2,tmplam;
	
	double[] x = new double[len]; // New variable values
	
	// Scale the step 
	for (sum=0.0,i=0;i<len;i++){
	    if(!isFixed[i]) // For fixed variables, direction = 0
		sum += direct[i]*direct[i];
	}	
	sum = Math.sqrt(sum);
	
	if (m_Debug)
	    System.err.println("fold:  "+Utils.doubleToString(fold,10,7)+"\n"+
			       "sum:  "+Utils.doubleToString(sum,10,7)+"\n"+
			       "stpmax:  "+Utils.doubleToString(stpmax,10,7));
	if (sum > stpmax){
	    for (i=0;i<len;i++)
		if(!isFixed[i])
		    direct[i] *= stpmax/sum;		
	}
	else
	    maxalam = stpmax/sum;
	
	// Compute initial rate of decrease, g'*d 
	m_Slope=0.0;
	for (i=0;i<len;i++){
	    x[i] = xold[i];
	    if(!isFixed[i])
		m_Slope += gradient[i]*direct[i];
	}
	
	if (m_Debug)
	    System.err.print("slope:  " + Utils.doubleToString(m_Slope,10,7)+ "\n");
	
	// Slope too small
	if(Math.abs(m_Slope)<=m_Zero){
	    if (m_Debug)
		System.err.println("Gradient and direction orthogonal -- "+
				   "Min. found with current fixed variables"+
				   " (or all variables fixed). Try to release"+
				   " some variables now.");
	    return x;
	}
	
	// Err: slope > 0
	if(m_Slope > m_Zero){
	    if(m_Debug)
		for(int h=0; h<x.length; h++)
		    System.err.println(h+": isFixed="+isFixed[h]+", x="+
				       x[h]+", grad="+gradient[h]+", direct="+
				       direct[h]);
	    throw new Exception("g'*p positive! -- Try to debug from here: line 327.");
	}
	
	// Compute LAMBDAmin and upper bound of lambda--alpha
	test=0.0;
	for(i=0;i<len;i++){	    
	    if(!isFixed[i]){// No need for fixed variables
		temp=Math.abs(direct[i])/Math.max(Math.abs(x[i]),1.0);
		if (temp > test) test=temp;
	    }
	}
	
	if(test>m_Zero) // Not converge
	    alamin = m_TOLX/test;
	else{
	    if (m_Debug)
		System.err.println("Zero directions for all free variables -- "+
				   "Min. found with current fixed variables"+
				   " (or all variables fixed). Try to release"+
				   " some variables now.");
	    return x;
	}
		
	// Check whether any non-working-set bounds are "binding"
	for(i=0;i<len;i++){
	    if(!isFixed[i]){// No need for fixed variables
		double alpi;
		if((direct[i]<-m_Epsilon) && !Double.isNaN(nwsBounds[0][i])){//Not feasible
		    alpi = (nwsBounds[0][i]-xold[i])/direct[i];
		    if(alpi <= m_Zero){ // Zero
			if (m_Debug)
			    System.err.println("Fix variable "+i+
					       " to lower bound "+ nwsBounds[0][i]+
					       " from value "+ xold[i]);
			x[i] = nwsBounds[0][i];
			isFixed[i]=true; // Fix this variable
			alpha = 0.0;
			nwsBounds[0][i]=Double.NaN; //Add cons. to working set
			wsBdsIndx.addElement(i);
		    }
		    else if(alpha > alpi){ // Fix one variable in one iteration
			alpha = alpi;
			fixedOne = i;
		    }			
		}
		else if((direct[i]>m_Epsilon) && !Double.isNaN(nwsBounds[1][i])){//Not feasible
		    alpi = (nwsBounds[1][i]-xold[i])/direct[i];
		    if(alpi <= m_Zero){ // Zero
			if (m_Debug)
			    System.err.println("Fix variable "+i+
					       " to upper bound "+ nwsBounds[1][i]+
					       " from value "+ xold[i]);
			x[i] = nwsBounds[1][i];
			isFixed[i]=true; // Fix this variable
			alpha = 0.0;
			nwsBounds[1][i]=Double.NaN; //Add cons. to working set
			wsBdsIndx.addElement(i);
		    }
		    else if(alpha > alpi){
			alpha = alpi;
			fixedOne = i;
		    }			
		}				
	    }
	}	
	
	if (m_Debug){
	    System.err.println("alamin: " + Utils.doubleToString(alamin,10,7));
	    System.err.println("alpha: " + Utils.doubleToString(alpha,10,7));
	}
	
	if(alpha <= m_Zero){ // Zero	   
	    m_IsZeroStep = true;
	    if (m_Debug)
		System.err.println("Alpha too small, try again");
	    return x;
	}
	
	alam = alpha; // Always try full feasible newton step 
	if(alam > 1.0)
	    alam = 1.0;
	
	// Iteration of one newton step, if necessary, backtracking is done
	double initF=fold, // Initial function value
	    hi=alam, lo=alam, newSlope=0, fhi=m_f, flo=m_f;// Variables used for beta condition
	double[] newGrad;  // Gradient on the new variable values
	
	kloop:
	for (k=0;;k++) {
	    if(m_Debug)
		System.err.println("\nLine search iteration: " + k);
	    
	    for (i=0;i<len;i++){
		if(!isFixed[i]){
		    x[i] = xold[i]+alam*direct[i];  // Compute xnew
		    if(!Double.isNaN(nwsBounds[0][i]) && (x[i]<nwsBounds[0][i])){    
			x[i] = nwsBounds[0][i]; //Rounding error	
		    }
		    else if(!Double.isNaN(nwsBounds[1][i]) && (x[i]>nwsBounds[1][i])){		
			x[i] = nwsBounds[1][i]; //Rounding error	
		    }
		}
	    }
	    
	    m_f = objectiveFunction(x);    // Compute fnew
	    if(Double.isNaN(m_f))
		throw new Exception("Objective function value is NaN!");
	
	    while(Double.isInfinite(m_f)){ // Avoid infinity
		if(m_Debug)
		    System.err.println("Too large m_f.  Shrink step by half.");
		alam *= 0.5; // Shrink by half
		if(alam <= m_Epsilon){
		    if(m_Debug)
			System.err.println("Wrong starting points, change them!");
		    return x;
		}
		
		for (i=0;i<len;i++)
		    if(!isFixed[i])
			x[i] = xold[i]+alam*direct[i]; 
		
		m_f = objectiveFunction(x); 
		if(Double.isNaN(m_f))
		    throw new Exception("Objective function value is NaN!");
	
		initF = Double.POSITIVE_INFINITY;
	    }
	    
	    if(m_Debug) {
		System.err.println("obj. function: " + 
				   Utils.doubleToString(m_f, 10, 7));
		System.err.println("threshold: " + 
				   Utils.doubleToString(fold+m_ALF*alam*m_Slope,10,7));
	    }
	    
	    if(m_f<=fold+m_ALF*alam*m_Slope){// Alpha condition: sufficient function decrease
		if(m_Debug)		
		    System.err.println("Sufficient function decrease (alpha condition): "); 
		newGrad = evaluateGradient(x);
		for(newSlope=0.0,i=0; i<len; i++)
		    if(!isFixed[i])
			newSlope += newGrad[i]*direct[i];

                if (m_Debug)
                  System.err.println("newSlope: " + newSlope);

		if(newSlope >= m_BETA*m_Slope){ // Beta condition: ensure pos. defnty.	
		    if(m_Debug)		
			System.err.println("Increasing derivatives (beta condition): "); 	

		    if((fixedOne!=-1) && (alam>=alpha)){ // Has bounds and over
			if(direct[fixedOne] > 0){
			    x[fixedOne] = nwsBounds[1][fixedOne]; // Avoid rounding error
			    nwsBounds[1][fixedOne]=Double.NaN; //Add cons. to working set
			}
			else{
			    x[fixedOne] = nwsBounds[0][fixedOne]; // Avoid rounding error
			    nwsBounds[0][fixedOne]=Double.NaN; //Add cons. to working set
			}
			
			if(m_Debug)
			    System.err.println("Fix variable "
					       +fixedOne+" to bound "+ x[fixedOne]+
					       " from value "+ xold[fixedOne]);
			isFixed[fixedOne]=true; // Fix the variable
			wsBdsIndx.addElement(fixedOne);
		    }		
		    return x;
		}
		else if(k==0){ // First time: increase alam 
		    // Search for the smallest value not complying with alpha condition
		    double upper = Math.min(alpha,maxalam); 
		    if(m_Debug)
			System.err.println("Alpha condition holds, increase alpha... ");
		    while(!((alam>=upper) || (m_f>fold+m_ALF*alam*m_Slope))){
			lo = alam;
			flo = m_f;
			alam *= 2.0;
			if(alam>=upper)  // Avoid rounding errors
			    alam=upper;

			for (i=0;i<len;i++)
			    if(!isFixed[i])
				x[i] = xold[i]+alam*direct[i];
			m_f = objectiveFunction(x);
			if(Double.isNaN(m_f))
			    throw new Exception("Objective function value is NaN!");
			
			newGrad = evaluateGradient(x);
			for(newSlope=0.0,i=0; i<len; i++)
			    if(!isFixed[i])
				newSlope += newGrad[i]*direct[i];
			
			if(newSlope >= m_BETA*m_Slope){
			    if (m_Debug)		
				System.err.println("Increasing derivatives (beta condition): \n"+
						   "newSlope = "+Utils.doubleToString(newSlope,10,7));
			    
			    if((fixedOne!=-1) && (alam>=alpha)){ // Has bounds and over
				if(direct[fixedOne] > 0){
				    x[fixedOne] = nwsBounds[1][fixedOne]; // Avoid rounding error
				    nwsBounds[1][fixedOne]=Double.NaN; //Add cons. to working set
				}
				else{
				    x[fixedOne] = nwsBounds[0][fixedOne]; // Avoid rounding error
				    nwsBounds[0][fixedOne]=Double.NaN; //Add cons. to working set
				}
				
				if(m_Debug)
				    System.err.println("Fix variable "
						       +fixedOne+" to bound "+ x[fixedOne]+
						       " from value "+ xold[fixedOne]);
				isFixed[fixedOne]=true; // Fix the variable
				wsBdsIndx.addElement(fixedOne);
			    }		 				    
			    return x;
			}
		    }
		    hi = alam;
		    fhi = m_f;			
		    break kloop;
		}
		else{
		    if(m_Debug)
			System.err.println("Alpha condition holds.");
		    hi = alam2; lo = alam; flo = m_f;
		    break kloop;
		}		    
	    }        
	    else if (alam < alamin) { // No feasible lambda found       
		if(initF<fold){ 
		    alam = Math.min(1.0,alpha);
		    for (i=0;i<len;i++)
			if(!isFixed[i])
			    x[i] = xold[i]+alam*direct[i]; //Still take Alpha
		    
		    if (m_Debug)
			System.err.println("No feasible lambda: still take"+
					   " alpha="+alam);
		    
		    if((fixedOne!=-1) && (alam>=alpha)){ // Has bounds and over
			if(direct[fixedOne] > 0){
			    x[fixedOne] = nwsBounds[1][fixedOne]; // Avoid rounding error
			    nwsBounds[1][fixedOne]=Double.NaN; //Add cons. to working set
			}
			else{
			    x[fixedOne] = nwsBounds[0][fixedOne]; // Avoid rounding error
			    nwsBounds[0][fixedOne]=Double.NaN; //Add cons. to working set
			}
			
			if(m_Debug)
			    System.err.println("Fix variable "
					       +fixedOne+" to bound "+ x[fixedOne]+
					       " from value "+ xold[fixedOne]);
			isFixed[fixedOne]=true; // Fix the variable
			wsBdsIndx.addElement(fixedOne);
		    }		 		    
		}
		else{   // Convergence on delta(x)
		    for(i=0;i<len;i++) 
			x[i]=xold[i];
		    m_f=fold;
		    if (m_Debug)
			System.err.println("Cannot find feasible lambda"); 
		}
		
		return x; 
	    }
	    else { // Backtracking by polynomial interpolation
		if(k==0){ // First time backtrack: quadratic interpolation
		    if(!Double.isInfinite(initF))
			initF = m_f;		    
		    // lambda = -g'(0)/(2*g''(0))
		    tmplam = -0.5*alam*m_Slope/((m_f-fold)/alam-m_Slope);
		}
		else {    // Subsequent backtrack: cubic interpolation 
		    rhs1 = m_f-fold-alam*m_Slope;
		    rhs2 = fhi-fold-alam2*m_Slope;
		    a=(rhs1/(alam*alam)-rhs2/(alam2*alam2))/(alam-alam2);
		    b=(-alam2*rhs1/(alam*alam)+alam*rhs2/(alam2*alam2))/(alam-alam2);
		    if (a == 0.0) tmplam = -m_Slope/(2.0*b);
		    else {
			disc=b*b-3.0*a*m_Slope;
			if (disc < 0.0) disc = 0.0;
			double numerator = -b+Math.sqrt(disc);
			if(numerator >= Double.MAX_VALUE){
			    numerator = Double.MAX_VALUE;
			    if (m_Debug)
				System.err.print("-b+sqrt(disc) too large! Set it to MAX_VALUE.");
			}
			tmplam=numerator/(3.0*a);
		    }
		    if (m_Debug)
			System.err.print("Cubic interpolation: \n" + 
					 "a:   " + Utils.doubleToString(a,10,7)+ "\n" +
					 "b:   " + Utils.doubleToString(b,10,7)+ "\n" +
					 "disc:   " + Utils.doubleToString(disc,10,7)+ "\n" +
					 "tmplam:   " + tmplam + "\n" +
					 "alam:   " + Utils.doubleToString(alam,10,7)+ "\n");	
		    if (tmplam>0.5*alam)
			tmplam=0.5*alam;             // lambda <= 0.5*lambda_old
		}
	    }
	    alam2=alam;
	    fhi=m_f;
	    alam=Math.max(tmplam,0.1*alam);          // lambda >= 0.1*lambda_old
	    
	    if(alam>alpha){
		throw new Exception("Sth. wrong in lnsrch:"+
				    "Lambda infeasible!(lambda="+alam+
				    ", alpha="+alpha+", upper="+tmplam+
				    "|"+(-alpha*m_Slope/(2.0*((m_f-fold)/alpha-m_Slope)))+
				    ", m_f="+m_f+", fold="+fold+
				    ", slope="+m_Slope);
	    }	    
	} // Endfor(k=0;;k++)
	
	// Quadratic interpolation between lamda values between lo and hi.
	// If cannot find a value satisfying beta condition, use lo.
	double ldiff = hi-lo, lincr;
	if(m_Debug)
	    System.err.println("Last stage of searching for beta condition (alam between "
			       +Utils.doubleToString(lo,10,7)+" and "
			       +Utils.doubleToString(hi,10,7)+")...\n"+
			       "Quadratic Interpolation(QI):\n"+
			       "Last newSlope = "+Utils.doubleToString(newSlope, 10, 7));
	
	while((newSlope<m_BETA*m_Slope) && (ldiff>=alamin)){
	    lincr = -0.5*newSlope*ldiff*ldiff/(fhi-flo-newSlope*ldiff);

	    if(m_Debug)
		System.err.println("fhi = "+fhi+"\n"+
				   "flo = "+flo+"\n"+
				   "ldiff = "+ldiff+"\n"+
				   "lincr (using QI) = "+lincr+"\n");
	    
	    if(lincr<0.2*ldiff) lincr=0.2*ldiff;
	    alam = lo+lincr;
	    if(alam >= hi){ // We cannot go beyond the bounds, so the best we can try is hi
	    	alam=hi;
		lincr=ldiff;
	    }
	    for (i=0;i<len;i++)
		if(!isFixed[i])
		    x[i] = xold[i]+alam*direct[i];
	    m_f = objectiveFunction(x);
	    if(Double.isNaN(m_f))
		throw new Exception("Objective function value is NaN!");
	
	    if(m_f>fold+m_ALF*alam*m_Slope){ 
		// Alpha condition fails, shrink lambda_upper
		ldiff = lincr;
		fhi = m_f;
	    }	    
	    else{ // Alpha condition holds	    
		newGrad = evaluateGradient(x);
		for(newSlope=0.0,i=0; i<len; i++)
		    if(!isFixed[i])
			newSlope += newGrad[i]*direct[i];
		
		if(newSlope < m_BETA*m_Slope){
		    // Beta condition fails, shrink lambda_lower
		    lo = alam;
		    ldiff -= lincr;
		    flo = m_f;
		}
	    }
	} 
	
	if(newSlope < m_BETA*m_Slope){ // Cannot satisfy beta condition, take lo
	    if(m_Debug)
		System.err.println("Beta condition cannot be satisfied, take alpha condition");
	    alam=lo;
	    for (i=0;i<len;i++)
		if(!isFixed[i])
		    x[i] = xold[i]+alam*direct[i];
	    m_f = flo;
	}
	else if(m_Debug)
	    System.err.println("Both alpha and beta conditions are satisfied. alam="
			       +Utils.doubleToString(alam,10,7));
	
	if((fixedOne!=-1) && (alam>=alpha)){ // Has bounds and over
	    if(direct[fixedOne] > 0){
		x[fixedOne] = nwsBounds[1][fixedOne]; // Avoid rounding error
		nwsBounds[1][fixedOne]=Double.NaN; //Add cons. to working set
	    }
	    else{
		x[fixedOne] = nwsBounds[0][fixedOne]; // Avoid rounding error
		nwsBounds[0][fixedOne]=Double.NaN; //Add cons. to working set
	    }
	    
	    if(m_Debug)
		System.err.println("Fix variable "
				   +fixedOne+" to bound "+ x[fixedOne]+
				   " from value "+ xold[fixedOne]);
	    isFixed[fixedOne]=true; // Fix the variable
	    wsBdsIndx.addElement(fixedOne);
	}
	
	return x;
    }
    
    /**
     * Main algorithm.  Descriptions see "Practical Optimization"
     *
     * @param initX initial point of x, assuming no value's on the bound!
     * @param constraints the bound constraints of each variable
     *                    constraints[0] is the lower bounds and 
     *                    constraints[1] is the upper bounds
     * @return the solution of x, null if number of iterations not enough
     * @throws Exception if an error occurs
     */
    public double[] findArgmin(double[] initX, double[][] constraints) 
	throws Exception{
	int l = initX.length;
	
	// Initially all variables are free, all bounds are constraints of
	// non-working-set constraints
	boolean[] isFixed = new boolean[l];
	double[][] nwsBounds = new double[2][l];
	// Record indice of fixed variables, simply for efficiency
	DynamicIntArray wsBdsIndx = new DynamicIntArray(constraints.length); 
	// Vectors used to record the variable indices to be freed 	
	DynamicIntArray toFree=null, oldToFree=null;	

	// Initial value of obj. function, gradient and inverse of the Hessian
	m_f = objectiveFunction(initX);
	if(Double.isNaN(m_f))
	    throw new Exception("Objective function value is NaN!");
	
	double sum=0;
	double[] grad=evaluateGradient(initX), oldGrad, oldX,
	    deltaGrad=new double[l], deltaX=new double[l],
	    direct = new double[l], x = new double[l];
	Matrix L = new Matrix(l, l);  // Lower triangle of Cholesky factor 
	double[] D = new double[l];   // Diagonal of Cholesky factor
	for(int i=0; i<l; i++){
          //	    L.setRow(i, new double[l]);  Not necessary
	    L.set(i,i,1.0);
	    D[i] = 1.0;
	    direct[i] = -grad[i];
	    sum += grad[i]*grad[i];
	    x[i] = initX[i];
	    nwsBounds[0][i] = constraints[0][i];
	    nwsBounds[1][i] = constraints[1][i];
	    isFixed[i] = false;
	}	
	double stpmax = m_STPMX*Math.max(Math.sqrt(sum), l);
	
	iterates:
	for(int step=0; step < m_MAXITS; step++){
	    if (m_Debug)
		System.err.println("\nIteration # " + step + ":");	    
	    
	    // Try at most one feasible newton step, i.e. 0<lamda<=alpha
	    oldX = x;
	    oldGrad = grad;
	    
	    // Also update grad
	    if (m_Debug)
		System.err.println("Line search ... ");
	    m_IsZeroStep = false;
	    x=lnsrch(x, grad, direct, stpmax, 
		     isFixed, nwsBounds, wsBdsIndx);
	    if (m_Debug)
		System.err.println("Line search finished.");
	    
	    if(m_IsZeroStep){ // Zero step, simply delete rows/cols of D and L
		for(int f=0; f<wsBdsIndx.size(); f++){
                    int[] idx = new int[1];
                    //		    int idx=wsBdsIndx.elementAt(f);
                    idx[0] = wsBdsIndx.elementAt(f);
                    L.setMatrix(idx, 0, l - 1, new Matrix(1, l));
                    //		    L.setRow(idx, new double[l]);
                    L.setMatrix(0, l - 1, idx, new Matrix(l, 1));
                    //		    L.setColumn(idx, new double[l]);
                    D[idx[0]] = 0.0;
                    //		    D[idx] = 0.0;
		}		
		grad = evaluateGradient(x);
		step--;
	    }
	    else{
		// Check converge on x
		boolean finish = false;
		double test=0.0;
		for(int h=0; h<l; h++){
		    deltaX[h] = x[h]-oldX[h];
		    double tmp=Math.abs(deltaX[h])/
			Math.max(Math.abs(x[h]), 1.0);
		    if(tmp > test) test = tmp;				    
		}
		if(test < m_Zero){
		    if (m_Debug)
			System.err.println("\nDeltaX converge: "+test);
		    finish = true;
		}
		
		// Check zero gradient	    
		grad = evaluateGradient(x);
		test=0.0;
		double denom=0.0, dxSq=0.0, dgSq=0.0, newlyBounded=0.0; 
		for(int g=0; g<l; g++){
		    if(!isFixed[g]){ 		   
			deltaGrad[g] = grad[g] - oldGrad[g];		  
			// Calculate the denominators			    
			denom += deltaX[g]*deltaGrad[g];
			dxSq += deltaX[g]*deltaX[g];
			dgSq += deltaGrad[g]*deltaGrad[g];
		    }
		    else // Only newly bounded variables will be non-zero
			newlyBounded +=  deltaX[g]*(grad[g]-oldGrad[g]);
		    
		    // Note: CANNOT use projected gradient for testing 
		    // convergence because of newly bounded variables
		    double tmp = Math.abs(grad[g])*
			Math.max(Math.abs(direct[g]),1.0)/
			Math.max(Math.abs(m_f),1.0);
		    if(tmp > test) test = tmp;	
		}
		
		if(test < m_Zero){
		    if (m_Debug)
			System.err.println("Gradient converge: "+test);
		    finish = true;
		}	    
		
		// dg'*dx could be < 0 using inexact lnsrch
		if(m_Debug)
		    System.err.println("dg'*dx="+(denom+newlyBounded));	
		// dg'*dx = 0
		if(Math.abs(denom+newlyBounded) < m_Zero)
		    finish = true;
		
		int size = wsBdsIndx.size();
		boolean isUpdate = true;  // Whether to update BFGS formula	    
		// Converge: check whether release any current constraints
		if(finish){
		    if (m_Debug)
			System.err.println("Test any release possible ...");
		    	
		    if(toFree != null)
			oldToFree = (DynamicIntArray)toFree.copy();
		    toFree = new DynamicIntArray(wsBdsIndx.size());
		    
		    for(int m=size-1; m>=0; m--){
			int index=wsBdsIndx.elementAt(m);
			double[] hessian = evaluateHessian(x, index);			
			double deltaL=0.0;
			if(hessian != null){
			    for(int mm=0; mm<hessian.length; mm++)
				if(!isFixed[mm]) // Free variable
				    deltaL += hessian[mm]*direct[mm];
			}
			
			// First and second order Lagrangian multiplier estimate
			// If user didn't provide Hessian, use first-order only
			double L1, L2;
			if(x[index] >= constraints[1][index]) // Upper bound
			    L1 = -grad[index];
			else if(x[index] <= constraints[0][index])// Lower bound
			    L1 = grad[index];
			else
			    throw new Exception("x["+index+"] not fixed on the"+
						" bounds where it should have been!");
			
			// L2 = L1 + deltaL
			L2 = L1 + deltaL;			
			if (m_Debug)
			    System.err.println("Variable "+index+
					       ": Lagrangian="+L1+"|"+L2);
			
			//Check validity of Lagrangian multiplier estimate
			boolean isConverge = 
			    (2.0*Math.abs(deltaL)) < Math.min(Math.abs(L1),
							      Math.abs(L2));  
			if((L1*L2>0.0) && isConverge){ //Same sign and converge: valid
			    if(L2 < 0.0){// Negative Lagrangian: feasible
				toFree.addElement(index);
				wsBdsIndx.removeElementAt(m);
				finish=false; // Not optimal, cannot finish
			    }
			}
			
			// Although hardly happen, better check it
			// If the first-order Lagrangian multiplier estimate is wrong,
			// avoid zigzagging
			if((hessian==null) && (toFree != null) && toFree.equal(oldToFree)) 
			    finish = true;           
		    }
		    
		    if(finish){// Min. found
			if (m_Debug)
			    System.err.println("Minimum found.");
			m_f = objectiveFunction(x);
			if(Double.isNaN(m_f))
			    throw new Exception("Objective function value is NaN!");	
			return x;
		    }
		    
		    // Free some variables
		    for(int mmm=0; mmm<toFree.size(); mmm++){
			int freeIndx=toFree.elementAt(mmm);
			isFixed[freeIndx] = false; // Free this variable
			if(x[freeIndx] <= constraints[0][freeIndx]){// Lower bound
			    nwsBounds[0][freeIndx] = constraints[0][freeIndx];
			    if (m_Debug)
				System.err.println("Free variable "+freeIndx+
						   " from bound "+ 
						   nwsBounds[0][freeIndx]);
			}
			else{ // Upper bound
			    nwsBounds[1][freeIndx] = constraints[1][freeIndx];
			    if (m_Debug)
				System.err.println("Free variable "+freeIndx+
						   " from bound "+ 
						   nwsBounds[1][freeIndx]);
			}			
			L.set(freeIndx, freeIndx, 1.0);
			D[freeIndx] = 1.0;
			isUpdate = false;			
		    }			
		}
		
		if(denom<Math.max(m_Zero*Math.sqrt(dxSq)*Math.sqrt(dgSq), m_Zero)){
		    if (m_Debug) 
			System.err.println("dg'*dx negative!");
		    isUpdate = false; // Do not update		    
		}		
		// If Hessian will be positive definite, update it
		if(isUpdate){
		    
		    // modify once: dg*dg'/(dg'*dx)	
		    double coeff = 1.0/denom; // 1/(dg'*dx)	
		    updateCholeskyFactor(L,D,deltaGrad,coeff,isFixed);
		    
		    // modify twice: g*g'/(g'*p)	
		    coeff = 1.0/m_Slope; // 1/(g'*p)
		    updateCholeskyFactor(L,D,oldGrad,coeff,isFixed);  		    
		}
	    }
	    
	    // Find new direction 
	    Matrix LD = new Matrix(l,l); // L*D
	    double[] b = new double[l];
	    
	    for(int k=0; k<l; k++){
		if(!isFixed[k])  b[k] = -grad[k];
		else             b[k] = 0.0;
		
		for(int j=k; j<l; j++){ // Lower triangle	
		    if(!isFixed[j] && !isFixed[k])
			LD.set(j, k, L.get(j,k)*D[k]);
		}		
	    }	    	
	    
	    // Solve (LD)*y = -g, where y=L'*direct
	    double[] LDIR = solveTriangle(LD, b, true, isFixed);	    
	    LD = null;
	    
	    for(int m=0; m<LDIR.length; m++){
		if(Double.isNaN(LDIR[m]))
		    throw new Exception("L*direct["+m+"] is NaN!"
					+"|-g="+b[m]+"|"+isFixed[m]
					+"|diag="+D[m]);
	    }
	    
	    // Solve L'*direct = y
	    direct = solveTriangle(L, LDIR, false, isFixed);
	    for(int m=0; m<direct.length; m++){
		if(Double.isNaN(direct[m]))
		    throw new Exception("direct is NaN!");
	    }
	    
	    //System.gc();
	}
	
	if(m_Debug)
	    System.err.println("Cannot find minimum"+
			       " -- too many interations!");
	m_X = x;
	return null;
    }
    
    /** 
     * Solve the linear equation of TX=B where T is a triangle matrix
     * It can be solved using back/forward substitution, with O(N^2) 
     * complexity
     * @param t the matrix T
     * @param b the vector B
     * @param isLower whether T is a lower or higher triangle matrix
     * @param isZero which row(s) of T are not used when solving the equation. 
     *               If it's null or all 'false', then every row is used.
     * @return the solution of X
     */     
    public static double[] solveTriangle(Matrix t, double[] b, 
					 boolean isLower, boolean[] isZero){
	int n = b.length; 
	double[] result = new double[n];
	if(isZero == null)
	    isZero = new boolean[n];
	
	if(isLower){ // lower triangle, forward-substitution
	    int j = 0;
	    while((j<n)&&isZero[j]){result[j]=0.0; j++;} // go to the first row
	    
	    if(j<n){
		result[j] = b[j]/t.get(j,j);
		
		for(; j<n; j++){
		    if(!isZero[j]){
			double numerator=b[j];
			for(int k=0; k<j; k++)
			    numerator -= t.get(j,k)*result[k];
			result[j] = numerator/t.get(j,j);
		    }
		    else 
			result[j] = 0.0;
		}
	    }
	}
	else{ // Upper triangle, back-substitution
	    int j=n-1;
	    while((j>=0)&&isZero[j]){result[j]=0.0; j--;} // go to the last row
	    
	    if(j>=0){
		result[j] = b[j]/t.get(j,j);
		
		for(; j>=0; j--){
		    if(!isZero[j]){
			double numerator=b[j];
			for(int k=j+1; k<n; k++)
			    numerator -= t.get(k,j)*result[k];
			result[j] = numerator/t.get(j,j);
		    }
		    else 
			result[j] = 0.0;
		}
	    }
	}
	
	return result;
    }

    /**
     * One rank update of the Cholesky factorization of B matrix in BFGS updates,
     * i.e. B = LDL', and B_{new} = LDL' + coeff*(vv') where L is a unit lower triangle
     * matrix and D is a diagonal matrix, and v is a vector.<br/>
     * When coeff > 0, we use C1 algorithm, and otherwise we use C2 algorithm described
     * in ``Methods for Modifying Matrix Factorizations'' 
     *
     * @param L the unit triangle matrix L
     * @param D the diagonal matrix D
     * @param v the update vector v
     * @param coeff the coeffcient of update
     * @param isFixed which variables are not to be updated
     */    
    protected void updateCholeskyFactor(Matrix L, double[] D, 
					double[] v, double coeff,
					boolean[] isFixed)
	throws Exception{
	double t, p, b;
	int n = v.length;
	double[] vp =  new double[n];	
	for (int i=0; i<v.length; i++)
	    if(!isFixed[i])
		vp[i]=v[i];
	    else
		vp[i]=0.0;
	
	if(coeff>0.0){
	    t = coeff;	    
	    for(int j=0; j<n; j++){		
		if(isFixed[j]) continue;		
		
		p = vp[j];
		double d=D[j], dbarj=d+t*p*p;
		D[j] = dbarj;
		
		b = p*t/dbarj;
		t *= d/dbarj;
		for(int r=j+1; r<n; r++){
		    if(!isFixed[r]){
			double l=L.get(r, j);
			vp[r] -= p*l;
			L.set(r, j, l+b*vp[r]);
		    }
		    else
		    	L.set(r, j, 0.0);
		}
	    }
	}
	else{
	    double[] P = solveTriangle(L, v, true, isFixed);	    
	    t = 0.0;
	    for(int i=0; i<n; i++)
		if(!isFixed[i])
		    t += P[i]*P[i]/D[i];	    	
	    
	    double sqrt=1.0+coeff*t;
	    sqrt = (sqrt<0.0)? 0.0 : Math.sqrt(sqrt);
	    
	    double alpha=coeff, sigma=coeff/(1.0+sqrt), rho, theta;
	    
	    for(int j=0; j<n; j++){
		if(isFixed[j]) continue;
		
		double d=D[j];
		p = P[j]*P[j]/d;
		theta = 1.0+sigma*p;
		t -= p; 
		if(t<0.0) t=0.0; // Rounding error

		double plus = sigma*sigma*p*t;
		if((j<n-1) && (plus <= m_Zero)) 
		    plus=m_Zero; // Avoid rounding error
		rho = theta*theta + plus;		
		D[j] = rho*d;
		
		if(Double.isNaN(D[j])){
		    throw new Exception("d["+j+"] NaN! P="+P[j]+",d="+d+
					",t="+t+",p="+p+",sigma="+sigma+
					",sclar="+coeff);
		}
		
		b=alpha*P[j]/(rho*d);
		alpha /= rho;
		rho = Math.sqrt(rho);
		double sigmaOld = sigma;
		sigma *= (1.0+rho)/(rho*(theta+rho));	 
		if((j<n-1) && 
		   (Double.isNaN(sigma) || Double.isInfinite(sigma)))
		    throw new Exception("sigma NaN/Inf! rho="+rho+
				       ",theta="+theta+",P["+j+"]="+
				       P[j]+",p="+p+",d="+d+",t="+t+
				       ",oldsigma="+sigmaOld);
		
		for(int r=j+1; r<n; r++){
		    if(!isFixed[r]){
			double l=L.get(r, j);
			vp[r] -= P[j]*l;
			L.set(r, j, l+b*vp[r]);
		    }
		    else
		    	L.set(r, j, 0.0);
		}
	    }
	}	
    }

  /**
   * Implements a simple dynamic array for ints.
   */
  protected class DynamicIntArray
    implements RevisionHandler {

    /** The int array. */
    private int[] m_Objects;

    /** The current size; */
    private int m_Size = 0;

    /** The capacity increment */
    private int m_CapacityIncrement = 1;
  
    /** The capacity multiplier. */
    private int m_CapacityMultiplier = 2;

    /**
     * Constructs a vector with the given capacity.
     *
     * @param capacity the vector's initial capacity
     */
    public DynamicIntArray(int capacity) {
      
      m_Objects = new int[capacity];
    }

    /**
     * Adds an element to this vector. Increases its
     * capacity if its not large enough.
     *
     * @param element the element to add
     */
    public final void addElement(int element) {
      
      if (m_Size == m_Objects.length) {
	int[] newObjects;
	newObjects = new int[m_CapacityMultiplier *
			     (m_Objects.length +
			      m_CapacityIncrement)];
	System.arraycopy(m_Objects, 0, newObjects, 0, m_Size);
	m_Objects = newObjects;
      }
      m_Objects[m_Size] = element;
      m_Size++;
    }

    /**
     * Produces a copy of this vector.
     *
     * @return the new vector
     */
    public final Object copy() {
      
      
      DynamicIntArray copy = new DynamicIntArray(m_Objects.length);
      
      copy.m_Size = m_Size;
      copy.m_CapacityIncrement = m_CapacityIncrement;
      copy.m_CapacityMultiplier = m_CapacityMultiplier;
      System.arraycopy(m_Objects, 0, copy.m_Objects, 0, m_Size);
      return copy;
    }

    /**
     * Returns the element at the given position.
     *
     * @param index the element's index
     * @return the element with the given index
     */
    public final int elementAt(int index) {
      
      return m_Objects[index];
    }
    
    /**
     * Check whether the two integer vectors equal to each other
     * Two integer vectors are equal if all the elements are the 
     * same, regardless of the order of the elements
     *
     * @param b another integer vector
     * @return whether they are equal
     */ 
    private boolean equal(DynamicIntArray b){
      if((b==null) || (size()!=b.size()))
	  return false;
      
      int size=size();
      
      // Only values matter, order does not matter
      int[] sorta=Utils.sort(m_Objects), sortb=Utils.sort(b.m_Objects);
      for(int j=0; j<size;j++)
	if(m_Objects[sorta[j]] != b.m_Objects[sortb[j]])
	  return false;
      
      return true;
    }

    /**
     * Deletes an element from this vector.
     *
     * @param index the index of the element to be deleted
     */
    public final void removeElementAt(int index) {
      
      System.arraycopy(m_Objects, index + 1, m_Objects, index, 
		       m_Size - index - 1);
      m_Size--;
    }

    /**
     * Removes all components from this vector and sets its 
     * size to zero. 
     */
    public final void removeAllElements() {
      
      m_Objects = new int[m_Objects.length];
      m_Size = 0;
    }

    /**
     * Returns the vector's current size.
     *
     * @return the vector's current size
     */
    public final int size() {
      
      return m_Size;
    }
    
    /**
     * Returns the revision string.
     * 
     * @return		the revision
     */
    public String getRevision() {
      return RevisionUtils.extract("$Revision: 8076 $");
    }
  }
}
